A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. We propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze certain ergodic properties of the corresponding absolute period foliation. It is a joint work with Igor Krichever and Sergey Lando.[-]